For the low rank tensor completion problem in the tensor train format, the function \(F\left({\cal X}\right)\)  is given by \(F\left({\cal X}\right)= {\cal P}_{\Omega }\left({\cal X}\right)- {\cal P}_{\Omega }\left({\cal T}\right)\) .
Notice that the update direction \(\xi \)  is also the solution of the following least squares problem [1]}:
\(\xi = \operatorname{argmin}_{\xi \in T_{{\cal X}}{\cal M}_{\mathbf {r}}}~\left\Vert   {\cal P}_{\Omega }{\cal P}_{T_{{\cal X}}{\cal M}_{\mathbf {r}}}\xi +{\cal P}_{\Omega }\left({\cal X}\right)-{\cal P}_{\Omega }\left({\cal T}\right)\right\Vert _{{\footnotesize {\mathsf {F}}}}^2.\) 
